Once you get to the point where notation is adding confusion instead of making things more clear, you should stop using it. If I needed to talk about this set I would write exactly what you said. something along the lines of.....

"Let $S$ denote the set of all maps from $\mathbb {R}^*_+$ to the set of all maps from $\mathbb {R}^*_+$ to $\mathbb {R}^*_+$."